Couplings used to connect shafts are required for many purposes. A brief list of examples could include frontwheel drives of automobiles, automatic assembly and processing machinery, certain machine tools, precision instruments, and automatic control devices. The literature dealing with this subject in general is extensive and a comprehensive list of references in which the subject is discussed in its various aspects may be found in a recent book by Dudita, Dudita Fl. Cuplaje mobile homocinetice. Editura Teknica, Bucharest, 1974, pp. 226-228.
To perform satisfactorily in many of the above and other applications, a coupling should be constructed in such a way that it readily permits changes in relative shaft orientation during operation and maintains the ratio of input to output shaft speed constant for all input shaft speeds and all relative shaft orientations lying in a certain range. Thus, a coupling should not only insure a constant speed ratio in various orientations, but also insure a constant speed ratio during a change from one relative orientation to another.
An example of a coupling in which the ratio of the speeds of rotation of a pair of shafts about their respective axes varies as the axes undergo relative time-varying reorientations is a Hooke coupling. Probably the most widely known use of the Hooke coupling is the use to which it is put as a universal joint which couples a pair of shaft members in the drive line of an automobile and other motor vehicles.
In many motor vehicles the axes of the shaft members coupled by the joint may be considered as being nearly collinear -- that is to say, an angle .theta. which the axis of one shaft member makes with the axis of the other shaft member is very small. Under these conditions, the ratio of the speeds of rotation, .gamma., of the shaft members about their respective axes is given, for a Hooke coupling, by the equation EQU .gamma. .apprxeq. 1 + (.theta./.omega.) .theta. sin .phi. cos .phi.
where .omega. is the angular speed of one of the shafts, .phi. is the angular displacement of this shaft about its axis of rotation with respect to a given reference plane in which both shaft axes are fixed, and .theta. is the time derivative of .theta.. From the foregoing equation, it can be seen that sufficiently large values of .theta. can give rise to appreciable fluctuations in the speed ratio, .gamma.. Such values of .theta. may be encountered when a vehicle traverses a bumpy road at relatively high speed. They may also be encountered in equipment using such couplings which is subject to high frequency vibrations such as equipment used in aircraft, ships, rockets and the like.